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<H1>incremental_all_shortest_paths_as_graph(+Graph, +DistanceArg, +SourceNode, +Modified, -Lengths, -SubGraph)</H1>
Incrementally computes all shortest paths from a single source to every reachable node given a list of modified edges
<DL>
<DT><EM>Graph</EM></DT>
<DD>a graph structure
</DD>
<DT><EM>DistanceArg</EM></DT>
<DD>which argument of EdgeData to use as distance (integer)
</DD>
<DT><EM>SourceNode</EM></DT>
<DD>source node number (integer)
</DD>
<DT><EM>Modified</EM></DT>
<DD>list of e/3 edge structures whose DistanceArg argument has been modified
</DD>
<DT><EM>Lengths</EM></DT>
<DD>array of numbers (minimum path lengths)
</DD>
<DT><EM>SubGraph</EM></DT>
<DD>a graph structure
</DD>
</DL>
<H2>Description</H2>
<P>
    Incrementally computes all shortest paths from the single source
    node SourceNode to every sink node which is reachable from it. The
    result is returned in the form of a sub-graph of the input graph,
    which contains all relevant edges.
</P><P>
    DistanceArg refers to the graph's EdgeData information that was
    specified when the graph was constructed. If EdgeData is a simple
    number, then DistanceArg should be 0 and EdgeData will be taken
    as the length of the edge. If EdgeData is a compound data structure,
    DistanceArg should be a number between 1 and the arity of that
    structure and determines which argument of the EdgeData structure
    will be interpreted as the edge's length. Important: the distance
    information in EdgeData must be a positive number.
</P><P>
    If DistanceArg is given as -1, then any EdgeData is ignored and
    the length of every edge is assumed to be equal to 1.
</P><P>
    SourceNode is the common starting point for the computed paths.
</P><P>
    Modified is the list of e/3 edge structures whose DistanceArg
    argument has been modified since the last computation for this
    SourceNode.
</P><P>
    The result is returned in the form of SubGraph, which is a
    sub-graph of the input Graph, containing the same nodes, but only
    those edges that are needed to construct the shortest paths for
    the given parameters.  SubGraph does not inherit the nodename
    information from Graph, this can be set explicitly if required.
</P><P>
    In addition, a Lengths array is returned, whose entries indicate
    the length of a shortest path from SourceNode to the corresponding
    sink node.  If there is no path from SourceNode to a sink node J,
    then Lengths[J] is uninstantiated.
</P>
<H4>Properties of the resulting SubGraph</H4>
<P>
    To generate an actual path from the resulting SubGraph, start from the
    sink node J, select one of its incoming edges (graph_get_incoming_edges/3)
    to find a predecessor node, and continue this process until the SourceNode
    is reached. Depending on the parameters, the following 3 cases can occur:
    <OL>
    <LI>Graph did not contain zero-length edges: in this
    case, SubGraph is cycle-free and shortest paths can be found by simply
    selecting arbitrary incoming edges until SourceNode is reached.
    <LI>Graph did contain zero-length edges: in this case,
    SubGraph may contain (zero-length) cycles which one may want to exclude
    when constructing paths.
    </P>
<H3>Modes and Determinism</H3><UL>
<LI>incremental_all_shortest_paths_as_graph(+, +, +, +, -, -) is det
</UL>
<H2>Examples</H2>
<PRE>
    ?- sample_graph(G), incremental_all_shortest_paths_as_graph(G, 0, 1, 0, L, E).
    G = graph(13, []([e(1, 6, 1), e(1, 2, 1), e(1, 7, 1)], [], ...)
    L = [](0, 1, 2, 3, 2, 1, 1, _326, _327, 2, 3, 3, 3)
    SG = graph(13, []([e(1, 7, 1), e(1, 6, 1), e(1, 2, 1)], [], ...)
    Yes (0.00s cpu)
    </PRE>
<H2>See Also</H2>
<A HREF="../../lib/graph_algorithms/possible_path-7.html">possible_path / 7</A>, <A HREF="../../lib/graph_algorithms/shortest_paths-4.html">shortest_paths / 4</A>, <A HREF="../../lib/graph_algorithms/single_pair_shortest_path-5.html">single_pair_shortest_path / 5</A>, <A HREF="../../lib/graph_algorithms/all_short_paths_as_edges-6.html">all_short_paths_as_edges / 6</A>, <A HREF="../../lib/graph_algorithms/all_short_paths_as_graph-6.html">all_short_paths_as_graph / 6</A>, <A HREF="../../lib/graph_algorithms/incremental_all_shortest_paths_as_edges-6.html">incremental_all_shortest_paths_as_edges / 6</A>, <A HREF="../../lib/graph_algorithms/single_pair_short_path-6.html">single_pair_short_path / 6</A>, <A HREF="../../lib/graph_algorithms/single_pair_all_short_paths_as_graph-7.html">single_pair_all_short_paths_as_graph / 7</A>
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